The functor \(A^{\min}\) on \(p\)-local spaces
It is know that there is a natural isomorphism \(H_*(\Omega \Sigma X,\mathbb F)\cong T(\tilde{H}_*(X,\mathbb F))\) for a field \(\mathbb F\) and that the tensor coalgebra \(T(V)\) has a further decomposition by using the Poincaré-Birkhoff-Witt Theorem, and one may expect that there exists a functorial non-trivial homotopy decomposition of \(\Omega \Sigma X\). The authors gave the finest such functorial decomposition for a \(p\)-torsion suspension \(X\) in their previous paper [On natural coalgebra decompositions of tensor algebras and loop suspensions. Mem. Am. Math. Soc. 701, 109 p. (2000; Zbl 0964.55012)]. In this paper, they generalize the previous results to loop suspensions of an arbitrary \(p\)-local path connected space \(X\). In particular, they show that there is a functorial decomposition \[ \displaystyle \Omega \Sigma X\simeq A^{\text{ min}}(X)\times \Big( \bigvee_{n=2}^{\infty}Q_n^{\text{ max}}(X)\Big) \] if \(X\) is a path connected \(p\)-local CW complex with several other useful results. To prove these results, they define a specific map \(\pi_X:\Big( \bigvee_{n=2}^{\infty}Q_n^{\text{ max}}(X)\Big)\to\Sigma X\) and show that \(\Omega \pi_X\) has a left homotopy inverse by using the calculations of algebraic James-Hopf maps. The key of the generalization is the fact in modular representation theory any idempotent in \((\mathbb Z/p)(\Sigma_n)\) lifts to an idempotent in \((\mathbb Z_{(p)})(\Sigma_n)\), where \(\Sigma_n\) stands for the symmetric group of \(n\)-letters.
scientific article; zbMATH DE number 5071437
scientific article